Warm-up

The Arctic Tern flies 24,000 miles every year.  How many miles is that per hour, per day, per week, and per month

11 Weird Facts About Math

Most people when they look at math, look at the everyday formulas. The ones we all tend to use but have you checked out some of the stranger, lesser known math facts?  These are ones you can share with the class and capture their attention.

1.  The Ham Sandwich Theorem from the area of topology.  This theorem states that it is possible to cut the sandwich so the two halves have exactly the same amounts of bread, ham, and cheese. Furthermore, the shape of the three together doesn't matter.

2. The biggest amount of money ever sued for.  One man from New York, sued New York City for two trillion trillion trillion dollars alleging he'd been bitten by a rabid dog while riding a bus and for having his picture taken without his permission. Two trillion, trillion, trillion is $2,000,000,000,000,000,000,000,000,000,000,000 which is more money than exists in the world.  His lawsuit was thrown out so no one had to pay.

3. Computers are becoming Grand Masters of various games.  For instance, the last person to win a chess game against a computer was in 2005.  Then in 2017, a computer won against the top rated Go player becoming a grand master.

4.  The number forty is the only written number whose letters are in alphabetical order.

5. Back in 1997, a sailor on the USS Yorktown entered a zero in the wrong place in a program and caused the program to divide by zero.  This resulted in the ship's computer shutting down and the ship  was unable to move for two hours.

6.  If you could fold a piece of paper in half 103 times, the paper would be the same thickness as the known universe.

7.  Shakespeare includes the word mathematics in the play, The taming of the Shrew, three times but its the only one of his plays to use the word.

8.  The word "Googol" was created by the 9 year old nephew of American Mathematician Edward Kasner to name the number 1 followed by 100 zeros.  The Google search engine was named after Googol thus making it a household word.

9. The word "Hundred" is actually a corruption of the Norse word hundredth which means 120 not 100 as we use it.

10. The word Calculus means pebbles in Greek and in Latin it means little stone or pebble.

11.The first mathematical formula for modeling epidemics came out in 1927.

I love odd facts about math.  Let me know what you think, I'd love to hear.  Have a great day.

Fantasy Football and Math

Fantasy Football is a big thing.  Adults have leagues where they set up teams, trade, and go through the math to figure out who they want to keep, who they want to trade for, and who they don't want on their team.

Some of this is determined by whether you use the standard scoring system or a point per reception (PPR).  If you use a standard scoring system, you prefer running backs but if you use the PPR system both wide receivers and running backs are about equal. If quarterbacks are worth four or six points, that will determine if you want a normal quarterback at the four points or someone who is much better for the six points.

Once you know exactly what is going on with points and such, you can then begin to put together your draft list.  Instead of looking at the number of points a player makes, look for how points he out scores other people doing the same position.  For fantasy football leagues, this is considered a much better indicator.

Now how do you take this information and put it into the math classroom?  Back in 2011, the New York Times published a lesson plan for the classroom on using Fantasy Football to teach math to students.  It provides links to resources to learn more about the topic so if you need to bring your students up to speed you can.

It has the class divide into pairs and each pair creates their own three person team of a quarterback, a running back, and a receiver using statistics to base their decisions on.  After naming their teams, they will share with the rest of the class, their choices and reasons why they chose those specific players.

The next step is to collect more data on each player to be used for matchups.  They would need to find the passing yards, touchdowns, and interceptions for the quarterback, passing yards allowed, passing touchdowns allowed, and interceptions for the NFL teams.  All this data can be typed into a spread sheet to organize it and make it easier to carry out the math for matchups and other things.

The article also includes a link to a Fantasy Points Scoring guide to help determine which matchups will create the most points.  Once they've gotten everything done, students are asked to monitor the games over the weekend so on the following Monday, they take the results and determine which team one.  The lesson plan also includes an extension.  The great thing about this lesson plan is that you the teacher do not have to know a lot about the how Fantasy Football works.

The above activity can be introduced using a Fantasy Football activity from Yummy Math.  This one has information on how to turn the stats into points and that information is used to answer question on players such as Tom Brady and Carson Wentz.  In addition, it also covers a different scoring system for wide receivers and tight ends. Towards the end, students are required to fill out a chart with players and their fantasy points along with creating an algebraic expression for total points.

The New York Times Activity will take several weeks to complete while the Yummy Math should take no more than two periods but the second gives a good introduction.  Let me know what you think, I'd love to hear.  Have a great day.

How Much More Does It Cost?

We are always trying to connect the math we teach in class with the real world.  I discovered an article that gave me an idea for  my Algebra I class when we start discussing percentages and percentage increases.

Imagine having students find the per pound cost of an expensive steak at their local store and then compare it with a pound of  Wagyu Beef that goes for around $1300 per pound.  That works out to almost a 200 percent difference.

Have them check out the cost of regular mushrooms versus the cost of one pound of Matsutake mushrooms that go for about $1000 per pound.  You can have students read about these mushrooms to find out why the cost is so expensive and  why its not likely to go down.

Then there are the Coffin Bay Oysters produced by the Coffin Bay Oyster Farm who charge $100 per oyster.  This is quite a bit more than the going price of $45 per dozen.  So if they compare the cost of one Coffin Bay Oyster with the cost of one oyster, they will see it again is a huge difference.

They could also compare a regular melon with the Yubarki King Melon which is only sold by auction due to its rarity and it has sold for between $6000 and $23,000 per melon. Imagine comparing that with the regular cost of a three pound cantaloupe.

Most items listed in the article can be compared to something people buy in the supermarket so that makes a really good homework assignment by having students go find the prices for similar items.  Once they have this information, they can reread the article to find the prices these luxury items sell for.

This data is perfect for a spread sheet because students can input all prices, then add the formula needed to calculate percent differences in the prices of the items,  or even add the formula to calculate the price increase between the usual price and the the luxury item goes for.  The information can then organized in some sort of graphical form.

If you want to add a bit more of a real world connection, explain these items are added to certain dishes to create more expensive versions that can go for a lot more. If you were a restaurant making a luxury version of an omelet, how much would the cost be of the materials such as the white truffle, caviar, mushrooms and moose cheese.  Add in the cost of coffee using the most expensive coffee beans and a piece of melon.  What do the raw ingredients cost and how much of a mark-up do you think should be put on it to arrive at a final price.  For these types of dishes, there will be a 300 to 500 percent markup because many of these items are increasing in cost.

They could also figure out the cost of dinner with a 4 ounce steak served with potatoes, a salad with balsamic vinegar, moose cheese, a bowl of red bird's nest soup,  a cup of coffee, and a piece of fruit either melon or watermelon.  Perhaps they should look at a dinner with fish instead.

Let them see how many different meals they could come up with, the cost of making the meal, and how much they could sell the meal for.  At the end, write a concluding statement noting the differences with mathematical details such as the percent difference between a regular dinner with steak, salad, soup, coffee and dessert vs the cost of a meal made with these expensive items.  They may not believe it but there are people out there who would pay for these meals.  Let me know what you think, I'd love to hear.  Have a great day.

Designing Math Projects.

It is easy to design projects for our students but it is much harder to design projects that are rich in mathematical ideas and interesting.  One of the go to projects I learned during student teaching was to have students design a game based on the math they studied but looking back, it really didn't do much for deeper thought.  Many choose board games modeled after the ones they knew with math problems instead of other things.

A math project should not be assigned just to assign something.  It should be designed around a driving question and a focus on a specific math concept because these projects need to utilize critical thinking skills.  It should also require the student to use 21st century skills, have both formative and summative learning, and include a variety of assessments.

Furthermore, students need to see the work as meaningful consequently they should have an opportunity to select how they will fulfill the project through choice.  They also need to use technology much as they would out in the work world and they should be able to really investigate the question so they find a new answer, a new product, or a new personal connection.

In addition, they should be asked to publicly share their final product because it motivates them to produce a higher quality result.  The presentation could be in the form of a gallery walk whether physical or digital, or place it in a blog, somewhere more people than just the teacher will see it. This also means they will know who their audience is.  In English, students are told to think of who their audience is and the same thing should be done in math.

At the end, students should include a reflection piece to discuss what they learned, how they learned it and what they accomplished by doing the project.

In the end be sure to tie projects to standards, create a rubric with clear expectations, have open-ended problems, look at both process and content when grading, determine the skills needed and how to provide scaffolding for the students who do not have them all, and place the accountability with the students.

These ideas can be used to either create your own or apply to any project you are looking at for your students. If the project does not have all of the above, maybe it won't provide as good a learning opportunity as it might.

Let me know what you think, I'd love to hear.  Have a great day.

Graphic Organizers For Problem Solving.

 I love using graphic organizers in my math classes, especially with the students who struggle because graphic organizers help students think.

Most graphic organizers help students separate the important from the not so important information.  Further more, graphic organizers can be used to help solve word problems.

If you have a graphic organizers divided into quarters with a small rhombus in the center, like the Frayer model graphic organizers,  you can label the top left one with "What do I need to find?"  The top right with "What do I need to do?".  The lower left is labeled "How I solved it." and the lower right has "My answer and how I know its right."  In the center, is the problem where they do the math.

This type of graphic organizer requires students to determine what is being asked so they don't just throw numbers together in the hopes they did it correctly.  It also offers the chance to try out different strategies to work the problem.  Once the answer is found, they have to explain how they found it and  why they know its correct.  They also a chance to make sure they've gone over the problem to make sure they got it all.

The big advantage this graphic organizer has over the traditional way of doing things is that it is not done in the same linear listing most students learn.  This graphic organizer has the understand the problem, devise a plan, carry out the plan, and check your work but not in the same order as the material is usually taught so students are less likely to misapply the process and can jot ideas down as they occur.

Students are able to work in which ever quadrant they need to without following the hierarchy normally taught.  Furthermore, the organization of this graphic organizer allows the teacher to check their thinking and their work, making it easier to identify where their thinking might be off or confused.  It is much easier to identify where student confusion lies.

Another advantage of this type of graphic organizer is that it makes it easier for students to answer open ended questions on state tests.   Usually low performing students usually are unable to show their work in more than one area while average students often have difficulty in organizing their thoughts and high achieving students skip steps in their work.  This graphic organizer helps students produce better work and achieve higher scores.

We use graphic organizers for most material in math but how many of us actually have students use one for problem solving?  I'd like to say I do but that isn't the truth.  This is one of those things I'll be using in my math classes for everyone because even the higher preforming students often struggle with problem solving because it requires more than the standard algorithm.

Let me know what you think, I'd love to hear, have a great day.

Warm-up

If you use 3000 gallons of water for every 180 showers you take, how many gallons of water do you use per shower?

Warm-up

If the faucet leaks one drop per second and in one year 3000 gallons of water have been lost, how much water is wasted each day?

Gone Camping

It's that time again when we take high school students out to camp.  Since its a new district, we'll only be out one night but before we got out, the students will work on community service projects.  Have a great weekend.

The Realities of Rounding.

We teach students to round through their years in school.  Most students learn the if the number is four or below, round down and if its five and above, round up.  These are great rules but they don't always work in real life.

In general banks and governments tend to round down if the value is no more than half a cent so $1.245 would round to $1.24 since its a half cent but if its $1.246, it would go up to $1.25. This is particularly true with sales tax, interest, and other situations.  They use the 5 and below rounds down while 6 and above rounds up.

On the other hand, if you calculate the number of cans of paint needed to cover your bedroom and it comes out as 4.1 gallons of paint, you will not round it down, you will round it up to make sure you have enough paint to complete the job.  Usually if its something physical like paint, cloth, tiles, it is accepted that you will round it up.  You always round up so you can finish the job.

I know if I buy yarn and the pattern says I need 3 skeins, I'll buy 4 because I usually mess up and run short.  If I don't buy that extra skein, I'll run out and the store will have run out of that particular dye lot.  When dealing with dye lots, they can vary but a little bit and you notice small differences.  Since I usually have left overs, I keep them and make stripped socks so I have some very unique results.

On the other hand, in accounting there are two different rounding methods they use. First they use decimal precision which tends to follow normal rules of rounding as long as the end result does not effect the representation of values.  If the final values are accurate, you are fine.  The other method is the rounding preferences one where you always round up as long as it does not effect the final values of the overall report.

There are circumstances where rounding down costs the company less money in the long run than rounding up.  If the company pays dividends out of their profit, if the amount rounds up, they end up paying more to its share holders while if it rounded down, it wouldn't effect them that way.

There are apparently some apps out there that will automatically round up your purchases to the next dollar, take the difference and place it in your savings account.  Some deposit the amounts daily so it automatically starts collecting interest while other apps wait until you've accumulated say $5 before they deposit.

There are some countries that have gotten rid of the one cent pieces which make things even more interesting.  If the purchase is paid for by credit card or check, the odd amount such as $1.89 is left alone but if you are paying cash, it will be rounded up to $1.90.  In Australia, they have rules where if its one or two cents, the amount is rounded down to the nearest tens amount but for three and four cents, the amount is rounded up to the nearest five amount.  For six and seven its rounded down to the nearest five and eight and nine get rounded up to the nearest tens. Australia did this to eliminate the one and two cent pieces from their currency.

However, when rounding Grade Point Averages, the rounding is only done to one place and it has to either go up or down one place.  For instance if your GPA is 3.32, it rounds to 3.4 but if the GPA is a 3.49, it would go up to 3.5 and a 3.99 or a 3.935 rounds to 3.9 because a 4.0 is only a 4.0 and you can't round up to it.

So now we've seen various ways rounding is used in real life and it doesn't always follow the rules we teach in school.  The method of rounding depends on the contextual situation and that varies wildly across the board.

Let me know what you think, I'd love to hear, have a great day.

How Many Tiles?

 The other night while I was reading a passage about people going into the building, the author mentioned the columns were covered with pictures made out of tile.  Columns and tiles.  A perfect real world problem using cylinders.  I have seen columns in the Anchorage airport decorated in tile portraying African animals.

Just think of the math involved in planning it.  One of the first things they had to know was the radius and the height of each column to find surface area, not volume.

The next thing to look at is the type of tile you plan to use.  Most times, they use a small one inch square tile to cover it although I've seen some a bit larger but not larger than 2 inches by 2 inches.  Since the tiles are in square inches, the surface area of the column needs to be in inches and not feet.  This is a good context to help students see why units need to be the same.  I already know a 1 foot by 1 foot rigid tile will not work as well on a cylinder as the small tiles.

If the columns are being covered in one color, then there isn't much left to be done but if a picture is being made out of tiles, one has to calculate the number of tiles needed in each color realizing the pictures are scaled smaller than the actual cylinder.  The easiest way would be to calculate the amount of tile for each color using a percent but that really isn't the easiest.  The easiest way is to draw the picture on graph paper, color it in, and count the number of each color while keeping in mind the scale factor to determine how many of each to order.

This leads to the next step of ordering each color.  The usual rule of thumb is to convert the number of tiles into boxes.  Usually boxes of tiles are labeled as covering so many square feet and your measurements were in square inches so you have to divide the square inches by 144 (12in by 12 in) to determine the square feet.  For projects, its always good to round up to whole numbers because things happen.  Tiles can break and its important to have extra.

One last step is to calculate the cost by multiplying the number of boxes of tile by the cost of each box so you know what you will be spending but its a decent idea to include sales tax for the actual price.  If you want to add one more element, have students include the price of the grout.

 So if say purple is represented by 18 squares and each square is 36 square inches, the total is 648 square inches.  Then 648/144 = 4.5 square feet.  If one box covers 10 square feet, you'd still have to buy one box because that is the minimum.  At $22.00 per box, you'd spend $22.00 x 1 box = $22.00 for the purple alone.

In one activity you've done surface area, conversions, rounding, sales tax, scale factor and total cost.   Let me know what you think, I'd love to hear.  Have a great day.

Social Media in the Math Classroom.

Most of my students have accounts on Facebook, Twitter, Instagram and other social media platforms and use them regularly.  I've seen student spend hours on them and wondered if there was a way to include social media in the math classroom.

Some of the reasons for allowing the use of social media are that it allows students to increase their collaboration because they can work in or out of school easily, it allows students to improve their communication skills because even the shyest can contribute, allows students to share resources easier, and it can better prepare them to join the workforce.

I know there are articles out there that explain how to use Facebook, Twitter, or Instagram in your classroom but I tend to hesitate because I'm not comfortable using such a main stream platform.  Over the years, I have used Schoology and Edmoto because I'd learned about them at conferences.  My high school students preferred Schoology over Edmoto but recently, I've relied on google classroom as my platform.  Although Google Classroom is not usually considered social media it is classified as a social network because members of the "class" do not have to be in the same physical location.  Now back to Twitter, Facebook, and Instagram.

The best way to set up a Twitter account for your classroom is to use a Group Tweet account which can be either public or private.  This means they do not need their own twitter account because this group account gives them access and the account keeps a permanent record of all tweets so students don't have to search through all previous hashtags.

A classroom Twitter account allows instant communication between the teacher and students or their parents or students can tweet their questions to the teacher for answers via Twitter or in class. Students can look back to see what they've done over the year.  Furthermore, you can have students summarize what they've learned or it could act as an exit ticket.  For homework, instead of using a worksheet, assign students different problems and they can twitter the answer.

It is also possible to create a Facebook account for your class only by using the privacy settings.  Students can document examples of math they find in the real world and include a short note on the mathematics of it.  Furthermore, Facebook allows teachers to record and store videos for use by the students later on.  In addition, the teacher can create polls for students to participate in,  or they can share educational content, or provide links to documents, or activities.

Even Instagram allows teachers to set up a private instagram account that is totally separate from their personal account.  Once the account its set up, you can post student work, feature a student of the week, share the steps used to complete a certain type of problem, share where you found math used in the real world, post reminders of due dates, answer questions, give a pop quiz in class where student work in groups to find the answer before posting their answers on Instagram.

Before doing any Twitter, Facebook or Instagram with my students, I'd first check my school's policy on use of social media, make sure letters are sent home to parents informing them of the planned use, and make sure you have signed permission to display the photos online.  These are the three most popular social media sites and it is possible, your students know a bit more about using them then you do.

Let me know what you think, I'd love to hear.  Have a great day.

Every Day Geometry

Yesterday, I attended one of the local churches.  It's been here for many years.  The inside caught my attention as its built in such a way as to be filled with geometric shapes.  I spent the time during the sermon looking at the various shapes.

Most of the opaque windows were filled with square pieces of bubbly glass.  I saw those windows and automatically though of how we usually show area that way.  Some of the windows had one to three squares in a darker yellow than the rest and my mind saw how fractions could represent the windows.  The fractions could either represent the number of darker colored squares, the lighter ones or light to dark.  So many possibilities.

In the front, behind the pulpit, there was an alcove with a parabolic top.  The figure on the left of the hand drawn ones represents the general shape.  I thought of either calculating the equation of the parabola at the top or figuring out how many degrees the arced area between the sides.  I would have drawn a three dimensional picture but I am not that great at drawing.

The raised ceiling in the middle was held up with a trapezoidal piece made of wood.  It appears to be an isosceles trapezoid broken up with smaller isosceles trapezoids except for the middle area which is made up of two right angle trapezoids. My eyes wanted to make them into isosceles triangles but the legs didn't meet so they weren't.

The sides of the raised ceilings were rectangles between each trapezoidal support beam but from an angle some looked more like parallelograms but they weren't.  It was only the angle that changed the perspective otherwise they were rectangles that were physically tilted inwards.   The flat ceiling at the sides was composed of rectangles formed by strips of wood. When the wooden strips met the wall, there was a triangle made up of more wood.  It looked like a scalene triangle.

The carpet also got some attention from me because it was made up of carpet squares.  Each carpenter square was composed of lots of parallel lines and the squares had been put down at 90 degree rotation so one set of lines ran up and down and the next ran left to right.

The walls were composed of planks of wood running lengthwise down the church from back to front on the sides below the windows before switching to boards running up and down perpendicular to the bottom ones.  In the front and pack it appeared all the boards ran up and down so they were perpendicular to the foundation.

I don't think I've been in another church that captured my attention as mathematically as this one did. I was thinking of all sorts of activities I could have students do with the architecture of the building.  I'll try to grab some pictures to share another time with people.

Let me know what you think,  I'd love to hear.  Have a great day.

Warm-up

A 2 Euro coin weighs 0.29 ounces.  How many will it take to make a pound?

Warm-up

If each penny weighs .088 ounce,  how many pennies will it take to make a pound?

"Help me please"

I've been back at work for just over a week.  There was the inservice for new teachers with a full day on the math books I've got and this week is the inservice for everyone.  Yesterday, I attended a one hour inservice on rich tasks and ways to do it so it is more effective.

At one point we looked at two different classrooms to see how they did the same task.  In one classroom, the teacher observed, asked students questions on their thinking, and eventually every group ended up writing down some very coherent explanations for how they got the answer even if the answer was wrong.

In the other class room, we saw the teacher give the students the problem before wandering around the room but the first group asked her how to do it so she tried to offer suggestions but ended up helping them before moving on.  Then the next group made the same request and by the third group, she was at the board showing them all how to do it so in essence she did it for them.  I recognized that scenario because my students at the last school were always asking me how to do it.

I know part of the reason they did that was because they'd had a middle school teacher who taught but  didn't do much if they played around, walked out of the classroom, etc. The other part is that I gave up after watching them basically do nothing for most of the period, especially when they all asked for the same help.  They wanted me to show them how to do it.

As part of the discussion between the middle school teacher and myself, we acknowledged that we do give in especially if the class is running out of time and we want them to "get" it before it ends.  Furthermore, when we try to institute the check with three classmates before asking for help, it doesn't seem to work because they ask three classmates "You got it done?" and the three answer nope.

Most students I deal with seem to think the ask three classmates means you see if they have the answer but they usually don't because as soon as they perceive they are stuck, they ask others who are stuck rather than initiating a conversation about the problem and why they are having trouble.  They haven't developed the persistence they need for these types of tasks.

I think many students honestly do not feel as if they can do the work especially if they have gaps in their learning.  These gaps can make them feel "stupid" and they withdraw but I think we need to change their view of themselves and failure in math.  I like the idea that when you do not get the correct answer you have failed instead you just found a way not to do it.  If we have students think of it as not having found the right way of doing it rather than failure, we might be able to turn their self image around and perhaps help build a better mindset.

It seems too often we praise the kids who do well but we don't always take time to praise the students who struggle for trying or for their persistence, or when they figure out where they made a mistake.  This type of praise can encourage a student to work longer and harder because they start to feel valued and see their choices are valid.

It has been suggested teachers often talk too long and in too much detail rather than letting the students try to figure things out by themselves.  Perhaps you could pass out sheets with the information so students can work together to figure it out, or talk for a few minutes before having students predict what the rest of the lesson is about, or have students do some sort of exploration before you teach the lesson so they can predict what the rest of the lesson is about.

It is a hard thing to counter learned helplessness and as a teacher with a soft heart, I hate when they ask me again and again to help them with it because when they've done nothing for half a period, I give in because I need them to get it.  Since I'm in a new district, I haven't met my new students yet and I don't know what attitudes they will have.

Let me know what you think, I'd love to hear.  Have a great day.

Learning Goals

I've just moved to a new school district. The evaluation tool is a different one from the ones I've used before so I'm having to learn a whole new system.   Using learning goals is one I'm getting evaluated on this year.  This is an interesting one because as I understand it, this one in the evaluation refers to what the student is supposed to learn in the section or chapter and how well they are progressing towards it.

Usually, I just post the objectives in the form of "I can" statements but I've never taken the time to discuss them with my students.  I'd write them up, students would copy them down but we didn't do anything with them, so now I need to.

It has been stated that teacher clarity is extremely important.  One part of teacher clarity is communicating what will be learned and how a student knows they've learned it. In other words, a learning goal is the mathematical concept they are learning and to name the way they will show they learned it. So there are three suggestions for improving the implementation of learning goals.

First, one should create learning goals or learning intentions complete with a balanced set of both procedural and higher level thinking skills for rating success.  One should consider whether the learning intention focuses on the most important concepts in the lesson or are they referring to activities or tasks the students will complete.  At least one of the success criteria should include an element which allows the student to explain how to do something or actually do it.  Another element of the success criteria should allow students to justify, model or explain at a higher level.

Based on this, there should be the learning goal focused on the most important concept in the lesson and at least two ways the student can show how they know they've learned it.  The learning goal might be "I can solve one step equations."  The two ways to show they've learned it is by showing they can solve the problems and then show someone else how to do it.

Second, create and use routines that help focus students on the success criteria as they go through the lesson.  One of the routines a teacher can use is "Take Stock".  This is where the teacher might ask what they have done so far, ask a student to describe what the success criteria means in their own words. and asking how what they've done connects to the criteria.  Another possible routine is to create an anchor chart with three statements on knowing they are track because.........., they haven't met the learning goal yet because........., and what do they still need to do to reach the goal.

Finally, use responses to determine the next step.  For instance, if students are almost there, you might provide whole group feed back to help them the last bit of the way. If there is a bit of a gap between where the students are and the learning goal, you might want to provide further instruction.  If you get responses that don't tell you enough, you might want to gather additional evidence before making a decision, and if students are on track, move on.

Part of this has shown me that I need to work on setting better learning goals because I have not supplied the second part or how they know they've met the goal.  Since I have to create a professional learning goal, I think I'm going to make learning goals and success criteria as my goal for this year.  I want to learn to write them better.

Let me know what you think, I'd love to hear.  Have a great day.

More Real Life Markups.

Today I have more examples of real life markups, some might surprise you but all will offer students a chance to calculate the markup and make comment on each problem.

Lets start with the popcorn sold at movie theaters.  Most people buy candy, popcorn, and a drink to take in and then complain how much it ran them.  The amount of popcorn needed for a small container runs about 35 cents but the theater sells it for $6.50.  That is quite a markup.  The reason they sell the popcorn for so much is to keep ticket prices lower.

On the other hand, an iPhoneX costs about $370 to make but sells for $999 or a bit under three times the amount it cost to make.  Not all phones have the same markup but for Apple, the cost for certain electronic parts is decreasing but the cost isn't.  Other electronics such as HDMI cables and printer cartridges also have good markups.  A HDMI cable costs around $4.00 to make but sell for around $6.00 while printer cartridges cost between $4 and $5 dollars to make yet sell for between $14 and $50 depending on the type of printer you have. The last printer cartridge I purchased ran $32 for black ink.

For people who love to give greeting cards, a birthday card usually costs $1 to $2 to make but if you buy a Hallmark card, that will cost you somewhere between $3 and $7 so the markup isn't quite as bad as for other products.  Look at designer jeans.  The ones sold by True Religion cost $50 to make but can retail for $300.

Even graphic calculators such as the TI-84 cost between $15 and $20 to make but retail for around $100. The actual selling price depends on if you catch it on sale.  On Amazon, it retails for about $107.  We all know college textbooks have a pretty good markup but most only cost $20 to print and yet colleges charge between $100 and $200 for the same textbook new.

Planning to get married?  One of the 25 yard Badgley Mischka wedding dress made out of a silk satin can cost $2500 to make but sells for over $7000. Diamond Rings are not quite as bad on the markup.  A one carat diamond ring can cost between $1000 and $3000 depending on cut and type of ring yet sell for at least $4000 at places such as Kay Jewelers or Jared.  Other products such as eye glasses only cost around $30 for the frame and lenses but can cost $300 or more for a brand name pair.

This information can be used for students to calculate the actual markup for each item and then they can comment on the markup.  At the same time, students can create a double bar graph for each item showing the cost and the final price before creating a second bar graph showing the actual markup.  It's not hard to turn this into a quick project with pictures for each item, information on each item's cost, selling price, and markup, and a conclusion.

There  are lots of other things out there with huge markups but these are the ones I found the cost to make and the selling price.  Let me know what you think, I'd love to hear.  Have a great day.

Grocery Store Markups.

 Usually when we teach mark-ups at school, we talk about mark-ups with fill in the blanks to complete table or its something we don't think about much but what if discussed mark-ups used in a grocery store to determine if that was the best place to buy an item.

Grocery stores are not the best place to buy non-grocery items such as light bulbs because these stores apply a 40 to 50 percent markup.  It is often cheaper to buy them at War-mart.

Then those meals the grocery store prepares like macaroni salad or meat loaf can have a 40 percent markup.  Deli products sold via the counter  such as sliced meats and cheeses also have a 50 percent markup.  In addition, most fresh butchered meats found in the store have a 60 percent mark-up.  These are not the frozen meats.

In addition, all those brand name boxes of presweetened cold cereal, the bright colored one the kids love have a 40 percent markup.  Over the counter name brands are often marked up 30 to 40 percent so generic brands are usually cheaper.

If you want to buy beauty supplies from a grocery store, they have a 50 percent markup while batteries have a 70 percent mark-up here.  If you decide to buy fresh produce out of season, grocery stores use a 75 percent mark-up.  The amazing thing is that most brand name spices have a mark-up of around 97 percent which means the store can purchase them for about half the cost.

Another item that has a 99 percent markup is that coffee you got from the coffee place just inside the door.  Coming in with a 100 percent markup are any baked good you buy that was made at the store.  Finally, is bottled water sold at a grocery store that comes with a 4000 percent markup.

Now what to do with all this information?  Let students go online to find the current prices of these items at the local grocery store or make it a homework assignment to go with parents and mark the prices down.   Once the students have prices for everything, they can figure out what the item cost the store based on the average mark-ups given.

From here students can take the two figure, wholesale and retail, students can create stacked bar graphs showing the two costs for each item so they can see how the wholesale and markup relate to each other.  At the end, have students write a paragraph to summarize everything.

This puts markup into perspective and uses things they often buy at the store.  Tomorrow, I'll share more markups for other types of items. I'll include the cost to make it and what it's sold for.  Let me know what you think, I'd love to hear.  Have a great day.

Soda Pop Sales

Most students I know, love their soda.  I've seen some go though 6 to 8 cans during a single school day.  I agree it is way too much and they spend around $1.25 per can so it adds up.  I'd love to let students check out statistics for soda and use that information to create graphs.

According to an article in Motley Fool, the number of cans of soda pop has been in a decline recently.  In fact, this decline began in 2004 and has continued every year since then.  The four brands that are the most popular are:  Coca Cola with an 18% share among sodas, Pepsi is in second place but it lost its contract with Arby's to Coca Cola.  Followed by Diet Coca Cola and Mountain Dew.

This site allows you to choose soda statistics either world wide or for a specific country and then it shows much of the information in graphic form so students have a chance to learn to interpret the information.

I looked at only the soda sold in the United States but it showed information for both carbonated and non-carbonated.  The nice thing is that I could choose one or the other if I wanted to look at more detailed information.

It states revenue for carbonated soda is $164,340 million which breaks down to about $499.91 per capita.  This gives the teacher a chance to check on the reasonableness of this.  It might sound reasonable but the reality is the many people such as myself so not drink soda so many people spend way more  to have it average that way.

The article goes on to say that the most revenue is generated in the United States.  This leads to a question of why might that be.  We know its our population but are students able to connect population with consumption?  The average per capita consumption is about 167.6 liters in the United States.

The first graph shows how the $164,340 million revenue is divide between home consumption and non-home consumption.  It breaks down to about double the amount is consumed out of the house as the amount consumed at home.  It also breaks down the revenue per capita into home consumption and that consumed outside of the home.  Although the revenue has been increasing the amount by volume has been decreasing especially for what is consumed at home and the average per capita volume is also decreasing, more is still being consumed at home.

In addition, you can click on a world map to see how much each country consumes.  What I found quite interesting is that Brazil is in second place for both carbonated and non-carbonated at just over $53,500 million as compared to the $245,000 million of the United States.  That is quite a big difference but Brazil has 209 million people compared to the United States with 345 million.

It would be interesting to do some comparisons between Brazil and the United States.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If the tallest tree in the world is now 380.3 feet tall and the second tallest tree is 327.5 feet, how much taller is the tallest tree than the second tallest tree in percents?

Warm-up

The tallest redwood ever found is 379.7 feet tall.  What is that in inches?

Two Uses Of Analogies.

Analogies are those wonderful groups of words that have relationships to each other such as ruler is to line as compass is to circle.  The only time I've ever really seen them used in on the GRE test if you want to get an advanced degree at a college.

The first way analogies are used in math is to help students develop their analytical skills because they often have to compare, contrast, or sequence.  Its understanding relationships between things.  For instance in the ruler is to a line as a compass is to a circle has a relationship of the instrument used to create the object.  A ruler is used to draw a line while the compass is used to create a circle.  If you have students actually verbalize the relationship, it improves both their vocabulary and their understanding of how words relate to each other.

Others such as ray:line::arc: circle shows the relationship of part to a whole because the ray is part of a line while the arc is part of a circle or 5:25::25:625 means that 5^2 = 25 and 25^2 = 625.  The first reads the ray is to the line as the arc is to the circle or five is to 25 as 25 is to 625.  Every analogy shows a relationship.

One way to help student see the relationships is after going over several as a group, before handing out a worksheet which has three of the four provided so they can figure out the last one.  The final step would be to have students create their own analogies from scratch.

It's fairly easy to find free materials for students to practice analogies.  It just takes a little bit of time and voila, you can find them even at Teachers pay Teachers.

The other way analogies can be used is to help students learn the order of things. These I'd neither seen nor heard of so these were a surprise.  These analogies demonstrate things in a visual way.  For instance when discussing the commutative property with students, we usually show them the A + B = B + A and tell them, it doesn't matter what order you add the number in, the answer is the same.  To show it we can discuss putting on  hat and coat vs shoes and socks.   For shoes and socks, we have to put the socks on first and then the shoes.  We can't change the order but when we put on a hat and coat, it doesn't matter whether the hat goes on first or the coat.  The order does not matter for the coat and hat and that is the same for the commutative property.

 As for associative, we show it as A + (B + C) = (A + B) + C.  I usually explain that it means we change who we associate with so in the first part B is associating with C while in the second part A is associating with A.  Another way of looking at it is (Light Blue) bucket = light (blue bucket). In the first part, the bucket is a light blue color while in the second part the blue bucket does not weigh much and this is an example of something that is not associative.  So students can see that not everything in that form is associative.

So there you have it, two ways you can use analogies to help improve mathematical understanding and vocabulary.  Let me know what you think, I'd love to hear.  Have a great day.

Acting Out is Good For Math

I stumbled across this article addressing the idea that if students act out certain things in math, it helps them do better in math.  I'm not talking about the one who act's out to disrupt the class rather its a way of bringing word problems to life.

There is research indicating students who act out word problems have a better understanding than those who just read the problem.  In addition, another study stated that elementary students who acted out word problems were less distracted than those only read the problem.

Researchers think this has to do with how babies learn language.  Often, parents will hold a bear up when they talk about a bear, or point to themselves when they talk about the mother or father so the infant has a visual to go with the words.  This held up when a group of first and second graders read a story.  One group just repeated key sentences while the other group added motion using toys or themselves.  After one week, the children reread the stories and the group who played with toys had better comprehension of the story.  In addition, they did a better job of making inferences about the material.

This type of activity was repeated with older students and word problems.  The group that acted out the word problem before visualizing it mentally, did a better job of answering the question because they were less likely to be distracted by irrelevant information.

It is recommended when using this technique to answer word problems, students first read over the problem before discussing which parts seem to be relevant and which parts can be ignored.  The second step is to act out the relevant parts of the word problem either by using props, toys, or physically moving around.  Acting out the word problem, brings the material to life so the can "see" it.  This is another way of visually representing the problem.

One way to use this is to assign each group a different problem.  Assign a certain amount of time for students to determine what needs to be acted out and then let them plan how to act it out.  Once everyone is ready, let one person read the problem out and have each group  present it to the rest of the class.  Give the class a chance to work the problem out before moving on to the next group.  For older students, let them explain why they decided certain information was irrelevant to solving the problem.

Another side to acting out word problems is it gets students moving around so they are not always sitting in one place all period.  let me know what you think, I'd love to hear.  Have a great day.

Old Math Magazine

The other day I renewed my membership to the National Council of Teachers of Mathematics.  I got a chance to explore the archives.  I discovered the archives contain copies of journals dating back to 1908.  Yes, you read that right, 1908.

The first issue of the Mathematics Teacher appeared in 1908 and was 44 pages in length.  One of the things in the magazine has not changed in all this time.

Under the article "What should be the aims in teaching Algebra and how to attain them."  One of the first comments is this:

"The first and most essential aim must be to enlist the interest of the pupil in the subject to the end that he will put forth per-sistent effort in its mastery."  

This sounds like teaching the student persistence so they work their way through the problems rather than giving out.  Later in the same article, the author made these statements.
"  From the nature of the subject there are three special aims that may well claim the attention of the teacher:

I. To increase the pupil's knowledge of mathematical short-hand.
2. To increase the pupil's knowledge of truth, mathematical truth, in preparing him for future work in college and technical schools.
3· To establish in the mind of the pupil the firm belief that he can of himself distinguish truth from error and to see to it that he forms the habit of proving all things.

Although the language is archaic, at least two of the goals are the same as today.  One is for the student can obtain a solid understanding of math so they do well in college and the other is that he can find his mistakes and is able to prove his work to be correct.  I'm n to sure what mathematical short-hand is but I'm thinking it means he has a good understanding of processes and methods of using symbols.

With in the article "The aims of teaching geometry", the author states the teacher " becomes better acquainted with his pupils and their individual peculiarities of mind. He learns what their difficulties really are, and hence can aid them more intelligently."  This sounds like something I got told years ago about getting to know your students well so you can notice where they are having problems before being able to help them.

There is a complete article devoted to the importance of getting students to check their work. The author indicates that it is important to have students check their work regularly and for the teacher to hold the students responsible for carrying out the checks.  By having students check their work, it helps increase accuracy of work and makes students feel better knowing they got the answers correct.

Many of the articles are actually summaries of papers read at the annual meeting but not all.  I had fun reading all these articles because they reflect the language and attitudes of the time but it was fascinating to see these authors stating things we are still discussing today such as persistence.

Let me know what you think, I'd love to hear.  Have a great day.

Guess This!

 I was reading an article on how visualization of concepts and vocabulary is important to understanding.  I know I usually just have students create a Frayer model for all new vocabulary words but I don't take it any further.  What if instead, we take 10 minutes every few days to have students practice their vocabulary by using games.

For all of these games, I would divide the class into smaller groups of three to four people who work together.  For the first one, I'd have people divided into groups of two.

One game is we've seen is for a person to place a word on their forehead and ask people questions about that word or have the other person give definitions of the word.  The person who has the word on their forehead has to guess it.

Another version would be for a person to draw a card and describe the vocabulary word to the others so they have to guess it.  They cannot use the word itself. For instance they might say "split it into smaller equal groups" for division or "The way you solve a problem following the rules" for order of operation.

There is always Pictionary where the student has to draw a picture of the vocabulary word.  For division they might draw 24 apples and draw rings around 6 apples so there are four groups.  For function, the student might draw a graph with a vertical line through it to prove it is a function.  White boards are great for this or iPads with drawing apps.

Instead of playing Pictionary, do charades so students can move around rather than sitting. For Order of Operations, they'd have to show three words, the first word they might arrange numbers into order or be a waitress, the last word they might imitate a doctor taking out a liver.  This last one can break up the monotony of sitting down.

No matter which game you select, you need to think about the vocabulary you use.  Not all vocabulary would work will in charades but might work better for describing.

The thing about games like these is they give students a chance to connect the vocabulary word with descriptions and sometimes actions.  Research indicates at least for elementary students, connecting the word to an action understand the vocabulary better than by just reading or writing it.  This extra activity helps the word become part of their vocabulary base which in turn adds to their base knowledge.

Let me know what you think, I'd love to hear. Have a great day.

Where In The World?

 I have been thinking of doing a "Where In The World...." series with mathematically important buildings. The buildings might be something such as the Great Pyramid of Giza or The Leaning Tower of Pisa.  Both are mathematically important. I might have the birthplace of a famous mathematician or the location of a building with a special geometric shape.

My idea is to show the location on a map such as google maps and then include three clues designed to help students guess the correct answer.


For the Great Pyramid of Giza, the location would be in the middle of Egypt.  The three clues might look like this.

1.  This is one of the seven wonders of the world.
2. 2 pi times the perimeter of the base = the hight
3. It is a square based pyramid.

This incorporates some ancient history into the math class.



Another dot might be in Vienna for the Eye Bank which is a trapezoid shaped building.

1.  I have four sides of which only one set of sides if parallel.
2.  I contain lots of greenery but not the kind you store in the credit union.
3.  I was named after what you see with.

We can't forget the Pentagon.  Its located in Washington, DC.  The map would show a dot on Washington, D.C.

The clues might read as follows.
1.  It has 5 sides.
2. It houses the Department of Defense.
3.  It was one of the buildings hit on 9/11.

I believe that doing this type of exercise will provide students with a cross curricular experience.  Although it may not utilize mathematical equations, it will show geometric shapes.

Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If you get 18 mpg when going 55 mph and it drops to 15mpg at 65 mph, what is the percent decrease?

Warm-up

If you get 25 mpg before the tune-up and 32 mpg after, what is the percent increase in mpg?

Mind Mapping in Math

Mind mapping by definition is a way to visually organize information for any topic.  It is usually associated with English because its a good way to organize the mechanics of a story but its not as frequently associated with math.

Mind mapping and other methods which include a visual element increase retention of the material.  Mind mapping tends to eliminate the linear organization found in notes and turn it into a more natural representation the brain finds easier to understand and retain.

As stated mind mapping helps memory retention, allows brainstorming and exploration of mathematical concepts, problems, and ideas.  Furthermore, it can help people learn and understand connections between the concept and the idea.  It allows notes to be taken quicker and improves communication.

Mind maps or mind plans make it easy to write down keywords or key ideas and then jot associated information off the main point or idea.  It reduces the amount of paper used so notes can appear on one page instead of being spread over several pages.  The nice thing is that there are apps out there that allow you to create mind maps complete with links to videos, files, websites, and documents so everything is in one place.  Mind maps can also be done by hand on a single piece of paper.

Furthermore, mind maps can be used to organize the tools needed to solve the problem.  When I say tools, I mean using the steps for solving a problem such as understanding the problem, devising a plan, carry out the plan, and check the answer.  Each step is the base with ideas flowing out to help the mind come up with the method and the answer.

In addition to using Mind maps for learning material, they can be used to plan out any projects assigned by the teacher because mind maps allows students to list everything needed to create a project from start to finish.  Furthermore, if students need to write a report to accompany their project or they have to write anything in Math, they can use a mind map to help organize their thoughts for the essay.

Mind maps can also be used to brainstorm possible ways to  solve equations or organize information given in word problems and ways to use the information to solve them. Furthermore, mind maps can also help figure out the information you want to include in any mathematical presentation.

This site has examples of mind maps for lower grades to give you a starting place to share with your students if you aren't sure how to use mind maps.  Let me know what you think, I'd love to hear.  Have a great day.

Advertising Costs

 One thing in math we never seem to discuss is the cost effectiveness of advertisements via television or radio.  We know the price goes up during the Super Bowl because it makes the news but what of other times a year.

Most advertisements on television run between 15 and 60 seconds with an average of 30 seconds in length.

The cost of all advertisements is based on CPM or cost per thousand viewers or how much it costs for 1000 people to see the ad.  The cost varies according to several factors, location being the most important one.  Each market has a different CPM cost.

A local commercial might cost $5 per 1000 viewers but for a nationally seen advertisement, a 30 second spot can run on average $115,000.  If the same ad is shown during the Super Bowl, it could run around $5.25 million.

The way to calculate the actual cost of the commercial is to take the number of viewers divided by 1000 and the result is multiplied by the CPM so if you have 500,000 people watching the add in Kansas City it would be 500,000/1000 = 500 x 14.36 or $7180 but the same ad in Los Angeles would be 500,000/1000 = 500 x $34.75 or $17,375.  Los Angeles costs more than New York City or Detroit.

The factors affecting the CPM include the age group and since those between 25 and 54 do the most watching, this is the group targeted by advertisers. They also look at the gender of the viewer because certain shows appeal to the males while others are preferred by females.  The more popular the television show, the more the network can ask.

Is the show on a broadcast station such as NBC or is it on a cable station such as TNT.  Usually cable channels can demand a bit more because those who watch cable usually have more money available.  Is the program live or is it recorded.  Usually live programs such as sports events cost more which is why the Super Bowl costs so much.  Furthermore, the time of day, time of year, and location.  The best time of the year is the fourth quarter because of the upcoming holidays while the period between 8 PM and 11 PM is when the most people watch.  Also it cost more to advertise in a large city area versus a small rural area.

Even national events such as presidential debate will drive up the cost of any advertisements.  Usually there are 4 two minute breaks in each half hour show.  This means there is limited space for advertisements.

In addition, there is a cost involved in creating the advertisement and I'll cover that another time. So advertisers must look at the price of advertising via television, perhaps radio, or print.  Today's entry gives an idea of how costs are calculated.

Let me know what you think, I'd love to hear.  Have a great day.